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Spectroscopy of Macromolecules For chemistry in general, spectroscopy is more often applied than, say, scattering, but…many other courses teach it better. Focus on these three: 1. Fluorescence 2. Circular Dichroism/ORD 3. NMR Reference: VanHolde (newer edition, “Physical Biochemistry” but the older editions are special, if you can still get them) 3 Danger: this is “what you need to know.” One could easily spend a whole semester on this alone.

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5.The operators for position, momentum, time and energy are as shown in this table. Position and time pass through unchanged; momentum and energy are filtered through the calculus. 7

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6.The wavefunction is determined from Schrodinger’s equation: The Hamiltonian operator H follows from a classical mechanics system worked out by Hamilton, where the classical operator was H = K + U, the sum of kinetic and potential energies. This equation ties the action of the energy operator to how the wave function responds. Compare this to Fick ’ s 2 nd law: Dd 2 c/dx 2 = - dc/dt. Like concentration, a wavefunction describes where something is. Time-dependent Schrodinger’s Eqn. 8

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Why Quantum Mechanics is Hard 9 Because we don’t study enough classical mechanics. Debye is said to have seen very little new in Quantum Mechanics; at least, the math is common to other stuff…if you have studied enough other stuff…which hardly any of us do! So far, we have seen that expectation values are similar to other averages we have computed: sum of probability times thing, divided by sum of probabilities to normalize. The wave function “sandwich” is new…and often associated with a particularly simple matrix mathematics. We also see that Schrodinger’s equation resembles Fick’s equation….which in turn resembles the heat flow equation all engineers learn.

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Eigenfunctions may permit us to replace calculus with multiplication. 10

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Basis sets let us create functions by summation of other functions. It sometimes works out that a set of eigenfunctions can be used to represent other functions. We say the desired function can be expanded in the set of eigenfunctions. Compare this to writing a vector in terms of unit vectors. Eigenfunctions are most useful when they are orthogonal and complete, meaning that they do not project on to one another and are sufficient to express arbitrary functions (again, compare the traditional unit vectors). Hermitian operators satisfy this. 11

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A simple example of an operator: kinetic energy only Classically, kinetic energy is E k =mv 2 /2 = p 2 /2m since p = mv. Check the table of Postulate 5 to get the QM analogue: The operator  is discussed in the Math Tuneup, as is its “square”,  2, the Laplacian. 12

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Time may not matter. Often, the time-dependence of  (q,t) is particularly boring; it might just be an oscillation that can be factored out from the positional, q-dependent part. This case is appropriate for stationary operators—ones with no time dependence. If you put the above equation into the full, time-dependent Schrodinger’s equation of Postulate 6, you get (see VanHolde—it is very easy) the simpler, time-independent Shrodinger’s equation, which is an Eigenequation with the particularly interesting and useful Eigenvalue, E. H-sigh equals E-sigh. Another equation for permanent memory storage. 13

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Van Holde treats all the usual simple systems. These notes derive from a modern VanHolde (VanHolde, Johnson & Ho). The authors march through systems you probably saw in PChem already, such as: –Free particle –Particle in box –Hydrogen atom –Approximate solutions (perturbation) –Small molecules (LCAO) In time, I hope this presentation will grow to cover some of those subjects, at least briefly. Meanwhile, I hope the foregoing made QM seem less weird. For now, we need a leap of reasonableness. 14

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Leap of Reasonableness QM is the explanation of things we know about atoms, going all the way back to Dalton’s Law of Multiple Proportions. QM explains why it’s CH 4, not C H It’s those stupid waves; together with boundary conditions (like the electron has to be somewhere) they give constructive & destructive interferences—nodes—which makes Chemistry an integer science. Think laser cavity, think wave trough, think booming bass at some positions in a room. For particle existence to be tied to wave amplitude is tantamount to saying: there are discrete energy levels associated with standing waves. 15

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Example: nearly harmonic vibration potential. Weird quantum-mechanical things: There is a zero-point energy. Discrete states are separated by near multiples of the zero point energy. More energy = more nodes in the wave function = fancier dancing by the electrons. If we consider an energy diagram for electronic energy levels, not vibrational levels, that fancier dancing corresponds to more complex orbitals—e.g., complicated f orbitals instead of simpler s orbitals. U r  E = h 16

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QM is useful to describe transitions between atomic or molecular energy states. Beer-Lambert Law (see Van Holde) Einstein-Planck Coefficients (to be added; meanwhile, I could weep, these authors make it so simple). Transition dipole Orientation of transition dipole } Let’s do these for now 17

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Earlier, we talked about how light grabs electrons and shakes them to produce scattering. We used the analogy of a boat floating on rough seas; it produces a little ripple as it bobs up and down. Molecules that absorb light do the grabbing instead. This stronger interaction is more like the water going over Niagara Falls. Absorption vs. Scattering 18

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Operators project. (this is a sentence, not a government program) Van Holde (Ch.8, p 373 et seq.) shows that a transition from one quantum state to another (absorption, emission) occurs when: 1)  E = h 2) the transition dipole  fi is finite and makes a strong projection on the electric field. Transition strength  19

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Transition! 20

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Think of the initial state as rolling along the runway. The wings catch some air and— voila!—transition to vertical acceleration. In this case, the initial and final vectors (oops, wavefunctions) are pictured as orthogonal. In QM transitions, this may or may not be allowed. The main thing is: the transition operator (wings) somehow couples one state (horizontal motion) into another (vertical motion). 21

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Selection rules: will it fly? Quantum mechanical transitions depend on: 1)how the transition dipole aligns to the electric field 2)obscure rules regarding how the field being operated on (initial wavefunction) relates to the new field (final wavefunction). Some transitions are “forbidden”—meaning they happen infrequently. 22

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Fluorescence solvent effects Example: is that protein aggregating? Put a hydrophobic probe in (pyrene?) and see if it “lights up” to indicate aggregation across the hydrophobic patch. Example: is that arborol self-assembling? Same solution to similar problem as above. + 29

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Fluorescence effects not always a tool, sometimes a nuisance. For FPR, the dye can get quenched and not undergo photobleaching. This can happen as a result of variables like pH or salt. Dye can self-quench if too concentrated. That is again a tool: calcein leakage test for vesicles. 30

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Fluorescence Machines Can be Simple Turner fluorometer, filter design. This is just an LS machine with filters. Detector filters 31

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Remember our timing diagram….Unlike light scattering, fluorescence is not instantaneous. It is a little bit more like somehow preparing radioactive elements: excited molecules spontaneously decide they have had enough of life in the fast lane and return to their ground state. Some tolerate the excited state longer than others  Scattering Fluorescence/ Phosphorescence Phosphorescence 

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Fluorescence Decay: yet another exponential for us to learn. Define: N = number of molecules in the excited state. The number dN decaying back to ground state is proportional to the number available in the excited state. at first later t N The number N is reported by the proportionate number dN that emit light. 33

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Quantum yield indicates environment. The minimum rate of decay would be the Einstein A value for spontaneous emission, which can be calculated from spectral width (VanHolde Eq ) The actual decay rate is higher, due to internal conversion, intersystem crossing, nonradiative transfer and any stimulated emission processes from interactions with stray photons. Quantum yield is defined as: q = A/k It is a sensitive indicator of the environment of the dye; q often increases when a dye binds to a molecule. Free dye may be quenched (see previous slides). 35

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Another way to think of quantum yield: ratio of (visible) photons in to photons out. Some dyes are very efficient light converters, with quantum yields approaching 100%. Fluorescein (which we often use in FPR) is so efficient that it is hard to know how efficient it is. About 90%-100% has been reported. High efficiency is a good thing for FPR: efficient light conversion means little heat production, less damage. absorbed emitted I 36

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Fluorescence Resonance Energy Transfer— There’s nothing to FRET about. The emitting group does not have to be the same as the absorbing group. Donor Acceptor h Donor + Acceptor Donor + Acceptor + h 37

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C-AMP dissociates protein kinase This is followed by labeling fluorescein to the catalytic subunit & rhodamine to the regulatory subunit. When kinase is in compact form, the fluorescein fluorescence is donated to the rhodamine acceptor. When C-AMP opens the structure up, fluorescein fluorescence grows. 42

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Polarization of fluorescence can be used to measure a new transport property, rotational diffusion. Light scattering is instantaneous, so draw a donut around the induced dipole and that’s what radiation pattern you will get. Fluorescence and, especially, phosphorescence are slower and the molecule may rotate, taking the (emission) dipole with it, before emission. Draw a donut around the dipole at the time of emission. 43

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I II II Absorption transition dipole Emission transition dipole   Vertical incident light  is in a molecular frame of Reference; this is not the rotation of the molecule but the different transition dipole of emission and absorption; for example, the molecule might distort a bit on absorption, so emission would be in some new direction. detector 44

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If molecule did not rotate (subscript zero)….. You can measure  by freezing out the motion—e.g.,put the molecule in some viscous solvent, cool it, etc. 45

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If the molecule does rotate… Then r will deviate from r o, tending more towards zero. (If r o was positive, r goes down). You can use this to estimate rotational diffusion coefficients, or at least to track how they are changing due to binding, adhesion or aggregation. Problem: it’s the rotation of the fluorphore, not necessarily the whole molecule. According to VanHolde, fluorometry is the only spectroscopic method that senses changes in molecular weight (e.g., due to aggregation). What do you think of that assertion? 46

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This is called steady state fluorescence polarization anisotropy. In this equation,  is the fluorescence decay time and  is a rotational correlation time (proportional to the inverse of rotational diffusion coefficient). You may recall that D r = kT/8  R 3 from our discussion of Hv DLS. So this means that 1/D r or  represent volume. 47

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Perrin plot: r -1 vs T How would you use this? From intercept, get r o, which is related to . Use slope to estimate  /V. If  is known from fluorescence decay (or maybe even literature) then obtain V. Plot not straight? Could be nonspherical shape, proteins changing with temperature (denaturing), binding to stuff as a function of temperature, etc. T/T/ 48

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The well-heeled & talented can do pulsed, polarized fluorescence… t r(t) 49 A recent (and unusual, I think) application of this is to early detection of cancer: abstract.cfm?URI=ao

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Van Holde Fig

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Everything you ever wanted to know about Fluorescence but were afraid to ask. Lakowicz: Principles of Fluorescence “Mostly drudgery.” A source of fluorphores: probes.invitrogen.com (worth linking again) 51

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Plane-polarized light can be thought of as two counter-rotating circularly polarized beams. 54

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Circular dichroism, the selective absorption of left or right-circularly polarized light, is used to characterize chirality. 55

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If the solvent absorbs heavily in the UV, CD is probably not possible. Then we use spectropolarimetry, a.k.a. Optical Rotatory Dispersion (ORD). 56 Have you seen this before?

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NMR Spectroscopy You could have a whole lifetime in NMR! The main application is to molecular fingerprinting: NMR is the primary tool that lets chemists know they have the right structure. Increasingly, it competes with X-ray diffraction for structural characterization. We will do just enough to introduce the variations that tell us something about polymer dynamics. This can be internal dynamics of bonds or, intriguingly, diffusion in complex systems. Ref: Van Holde, Ch. 12 and articles containing the word, DOSY 57

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NMR is about nuclear spin flips +1/2 -1/2 Spin quantum numbers are multiples of ½. For protons, the choices are + ½ and –½. We may just call this + and -. How do we know? Because we can see energy get absorbed to change the spins, but only when a magnetic field is applied. The phenomenon is obviously in the quantum domain, but it is reminiscent of classical spin of a charge, which generates a magnetic dipole. 59

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Nuclear spins in a magnetic field can be compared to actual spin in a gravitational field. Play with gyroscope: if we want to deflect its motion, what is the right stimulus? 61

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NMR = Low-energy spectroscopy Today’s biggest NMR is about 900 MHz The energies associated with nuclear spin flips (from + to -) are much less than the thermal energy, kT: E + - E -  0. This is true even in high magnetic fields Unlike vibrational & electronic spectroscopy, most molecules are not in the ground state. In fact, only a tiny excess of molecules is in the ground state. 62

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Net magnetization of lots of spins The condition +1/2 low energy -1/2 high energy H There is no coherence among spins; they just rotate in random phase. If you count very carefully, you’ll see there are more low-E spins than high-E spins. Note: there is a good chance the signs are backwards—who cares? The slight excess of low-E spins sums to produce a net magnetization (heavy arrow). Because there are lots of spins, randomly phased, this heavy arrow really sums up parallel to the z-axis. M = net magetization 63

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If NMR were still like this… It would still be the most important weapon in the chemist’s arsenal. Even at low magnetic fields, the “fingerprinting” of simple molecules is easily achieved. 65

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Modern Fourier Transform NMR is even better. Very strong superconducting magnets. The magnet is not swept, but sometimes spatial gradients in field are arranged. “White light” excitation through RF pulses excites all nuclei at once. Relaxation experiments not different in basic concept from fluorescence. Imaging possible; changes in “image” can be related to diffusion. 66

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What is the Fourier Transform of “Ping”? t I  I A short pulse corresponds to a region of nearly constant intensity. Just one example of inverse relationships. Here’s another: in optics, if you want to image a tiny spot in a microscope, you need to gather scattered light off of it through a very wide angle objective (high numerical aperture or low “f number”). 67

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We need a bit more about how it works. Pulses are delivered along the x-direction. It’s a coherent pulse, with the energy and duration designed to bring half the spins in the ground state into the excited state: no net magnetization along z axis. It does this with phase coherence, so M rotates in the the x-y plane. A coil in the x-direction can sense (“acquire”) the oscillating projection of M onto that direction. These pulses, and the subsequent acquisition are much faster than sweeping the field as in a CW instrument. For almost all applications, FT-NMR has taken over. z z x Pulse y x y 69

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Detection Merry-go-round. Kids have to hang arm-to-arm so the last one can ring bell. Maybe they even hold hands so someone can reach out to do it!. 70 /watch?v=xjqBsan9De8

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Relax—the nuclei do! Over time, some of the spins in the high-energy state fall back to the low energy state. This enthalpic process happens over a characteristic time, T 1. If no one is on the merry-go-round, the bell never rings. Over another time, kids get sick of cooperating. They lose phase. Although a given group (say, those wearing red shirts) goes around as often as they ever did, they can no longer reach out and touch the bell. This entropic dephasing time is called T 2. 71

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FID = Free Induction Decay Why again do they oscillate? Why again do they decay? FID for one nucleus. FID for two nuclei (very different frequencies). 73

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You have to FT your FID. Frequencies of oscillation give NMR frequencies. You get strengths at those frequencies, too! Rapidity of decay controls width of peaks. 74 An efficient algorithm for doing this was invented ca by Cooley-Tukey and is called Fast Fourier Transformation. “Window” functions should be applied to limit the effects of sudden start/stop of acquired dta. (http://en.wikipedia.org/wiki/Cooley%E2%80%93 Tukey_FFT_algorithm)

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Pulse terminology is easiest in the rotating frame: as you spin at the Larmor frequency, look up, out or down. The pulse that was used to bring half the ground state spins into the excited state, resulting in coherent oscillation of M in the x-y plane is called a 90 o pulse. z z x 90 o Pulse y x y 75

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180 o Pulse A longer and/or more energetic pulse will take the excess of protons in the ground state and create a similar excess in the excited state. z z x 180 o Pulse y x y 76

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Measuring T 1 : Inversion Recovery Pulse Sequence* *Just try typing that into Google. t  FT  Spins are dunked upside-down, given a time,   to right themselves by spin-lattice relaxation. Some spin-spin occurs, too, but doesn’t matter because spins are refocused with the 90 o pulse and measured. The signal depends on the time,   Waterfall plot: you actually see something like this for all of the peaks in the spectrum.  IoIo IoIo  T1T1 I max /e I max 77

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T 1, T 2 can be used to assess crystallinity, glass transitions. 80 T 1 >= T 2 Fast-moving nuclei (left side of plot) relax by real motion. Slow nuclei (crystals) relax by spin dephasing. The motion time of the nucleus is called the correlation time,  c.

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The DOSY Dog Track: NMR is for the dogs. First, imagine that T 2 dephasing is nil. At the 90 pulse, all the dogs (spins) leave the starting gate and go along the track however far they go. Then the “come” command (180 pulse) is issued. Dogs run back as fast as they ran forward, and bark as the cross the line: echo! 81 Thanks to Frank Blum

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Wet Dog Track Suppose now the track is wet, with some lanes being particularly slow. Dogs come back at the same time! Unless they cross lanes! Then, dog that went out on the slow lane might return on a fast lane and be the first to bark. Dog that went out fast can return slow and be the last to bark. The spread in the time from first bark to last bark contains information about how long it takes dogs to cross tracks from slow to fast: Diffusion! 82

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PFGNMR = DOSY t  Echo Gradient pulses: inhomogeneous magnetic field = lanes that are faster or slower (spread of Larmor frequencies). The effect of the gradients is to distribute the echo over a wider time, which lowers the maximum echo. The echo amplitude depends on a parameter Q, determined by gradient strength, duration, and timing a bit similar to DLS or FPR. 83

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PFGNMR-DOSY Pros & Cons Pros No labeling! Chemical specificity anyway (DOSY often keeps the identities of the protons): measure diffusion of everything in a mixture. Did I mention no labeling? Works well precisely where DLS doesn’t: small diffusers. (FPR works for small diffusers—provided the dye doesn’t mess up the molecule being studied). Data are very quiet. Effectively no baseline issues. Did I mention no labeling? Cons Slow diffusers: T 2 might wipe out your signal before molecules diffuse much. Does not span the wide range of times and diffusers that DLS or even FPR does. Struggles with convection issues. Diffusion can be over a very short range of space. Is time-limited, not distance limited. Software for it still sucks. 84